The image you've uploaded contains a math problem about finding the area of a polygon using its vertices. It looks like the problem is asking for the formula to compute the area based on the vertices in counterclockwise order, and there's a specific set of choices provided for the formula. Also, part (c) of the question involves calculating the area of a pentagon using specific vertex coordinates.

Let's first break down the question:

Part (b)

This part describes a polygon with vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ in counterclockwise order. It asks you to find the area of the polygon using the following formula:

$A = \frac{1}{2} \left[ (x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + \cdots + (x_{n-1} y_n - x_n y_{n-1}) + (x_n y_1 - x_1 y_n) \right]$

This formula is known as the shoelace theorem (or Gauss's area formula). The formula sums up the terms formed by the products of the coordinates, following the counterclockwise order of the vertices.

Part (c)

For part (c), you are tasked with calculating the area of a pentagon given the vertices $(0, 0), (1, 3), (1, 2), (0, 1), (-3, 1)$. You would plug these coordinates into the shoelace formula to compute the area.

Let me know if you'd like me to go step-by-step to calculate the area of the pentagon or if you have further questions about the theory!